MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  flimclslem Structured version   Visualization version   GIF version
Theorem flimclslem 23495
Description: Lemma for flimcls 23496. (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Stefan O'Rear, 6-Aug-2015.)
Hypothesis
Ref Expression
flimcls.2 𝐹 = (𝑋filGen(fiβ€˜(((neiβ€˜π½)β€˜{𝐴}) βˆͺ {𝑆})))
Assertion
Ref Expression
flimclslem ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ (𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝑆 ∈ 𝐹 ∧ 𝐴 ∈ (𝐽 fLim 𝐹)))
Proof of Theorem flimclslem
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 flimcls.2 . . 3 𝐹 = (𝑋filGen(fiβ€˜(((neiβ€˜π½)β€˜{𝐴}) βˆͺ {𝑆})))
2 topontop 22422 . . . . . . . . 9 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝐽 ∈ Top)
323ad2ant1 1133 . . . . . . . 8 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ 𝐽 ∈ Top)
4 eqid 2732 . . . . . . . . 9 βˆͺ 𝐽 = βˆͺ 𝐽
54neisspw 22618 . . . . . . . 8 (𝐽 ∈ Top β†’ ((neiβ€˜π½)β€˜{𝐴}) βŠ† 𝒫 βˆͺ 𝐽)
63, 5syl 17 . . . . . . 7 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ ((neiβ€˜π½)β€˜{𝐴}) βŠ† 𝒫 βˆͺ 𝐽)
7 toponuni 22423 . . . . . . . . 9 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 = βˆͺ 𝐽)
873ad2ant1 1133 . . . . . . . 8 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ 𝑋 = βˆͺ 𝐽)
98pweqd 4619 . . . . . . 7 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ 𝒫 𝑋 = 𝒫 βˆͺ 𝐽)
106, 9sseqtrrd 4023 . . . . . 6 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ ((neiβ€˜π½)β€˜{𝐴}) βŠ† 𝒫 𝑋)
11 toponmax 22435 . . . . . . . . . 10 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 ∈ 𝐽)
12 elpw2g 5344 . . . . . . . . . 10 (𝑋 ∈ 𝐽 β†’ (𝑆 ∈ 𝒫 𝑋 ↔ 𝑆 βŠ† 𝑋))
1311, 12syl 17 . . . . . . . . 9 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ (𝑆 ∈ 𝒫 𝑋 ↔ 𝑆 βŠ† 𝑋))
1413biimpar 478 . . . . . . . 8 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋) β†’ 𝑆 ∈ 𝒫 𝑋)
15143adant3 1132 . . . . . . 7 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ 𝑆 ∈ 𝒫 𝑋)
1615snssd 4812 . . . . . 6 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ {𝑆} βŠ† 𝒫 𝑋)
1710, 16unssd 4186 . . . . 5 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ (((neiβ€˜π½)β€˜{𝐴}) βˆͺ {𝑆}) βŠ† 𝒫 𝑋)
18 ssun2 4173 . . . . . 6 {𝑆} βŠ† (((neiβ€˜π½)β€˜{𝐴}) βˆͺ {𝑆})
19113ad2ant1 1133 . . . . . . . 8 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ 𝑋 ∈ 𝐽)
20 simp2 1137 . . . . . . . 8 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ 𝑆 βŠ† 𝑋)
2119, 20ssexd 5324 . . . . . . 7 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ 𝑆 ∈ V)
2221snn0d 4779 . . . . . 6 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ {𝑆} β‰  βˆ…)
23 ssn0 4400 . . . . . 6 (({𝑆} βŠ† (((neiβ€˜π½)β€˜{𝐴}) βˆͺ {𝑆}) ∧ {𝑆} β‰  βˆ…) β†’ (((neiβ€˜π½)β€˜{𝐴}) βˆͺ {𝑆}) β‰  βˆ…)
2418, 22, 23sylancr 587 . . . . 5 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ (((neiβ€˜π½)β€˜{𝐴}) βˆͺ {𝑆}) β‰  βˆ…)
2520, 8sseqtrd 4022 . . . . . . . . . . 11 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ 𝑆 βŠ† βˆͺ 𝐽)
26 simp3 1138 . . . . . . . . . . 11 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†))
274neindisj 22628 . . . . . . . . . . . 12 (((𝐽 ∈ Top ∧ 𝑆 βŠ† βˆͺ 𝐽) ∧ (𝐴 ∈ ((clsβ€˜π½)β€˜π‘†) ∧ π‘₯ ∈ ((neiβ€˜π½)β€˜{𝐴}))) β†’ (π‘₯ ∩ 𝑆) β‰  βˆ…)
2827expr 457 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝑆 βŠ† βˆͺ 𝐽) ∧ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ (π‘₯ ∈ ((neiβ€˜π½)β€˜{𝐴}) β†’ (π‘₯ ∩ 𝑆) β‰  βˆ…))
293, 25, 26, 28syl21anc 836 . . . . . . . . . 10 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ (π‘₯ ∈ ((neiβ€˜π½)β€˜{𝐴}) β†’ (π‘₯ ∩ 𝑆) β‰  βˆ…))
3029imp 407 . . . . . . . . 9 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†)) ∧ π‘₯ ∈ ((neiβ€˜π½)β€˜{𝐴})) β†’ (π‘₯ ∩ 𝑆) β‰  βˆ…)
31 elsni 4645 . . . . . . . . . . 11 (𝑦 ∈ {𝑆} β†’ 𝑦 = 𝑆)
3231ineq2d 4212 . . . . . . . . . 10 (𝑦 ∈ {𝑆} β†’ (π‘₯ ∩ 𝑦) = (π‘₯ ∩ 𝑆))
3332neeq1d 3000 . . . . . . . . 9 (𝑦 ∈ {𝑆} β†’ ((π‘₯ ∩ 𝑦) β‰  βˆ… ↔ (π‘₯ ∩ 𝑆) β‰  βˆ…))
3430, 33syl5ibrcom 246 . . . . . . . 8 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†)) ∧ π‘₯ ∈ ((neiβ€˜π½)β€˜{𝐴})) β†’ (𝑦 ∈ {𝑆} β†’ (π‘₯ ∩ 𝑦) β‰  βˆ…))
3534ralrimiv 3145 . . . . . . 7 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†)) ∧ π‘₯ ∈ ((neiβ€˜π½)β€˜{𝐴})) β†’ βˆ€π‘¦ ∈ {𝑆} (π‘₯ ∩ 𝑦) β‰  βˆ…)
3635ralrimiva 3146 . . . . . 6 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ βˆ€π‘₯ ∈ ((neiβ€˜π½)β€˜{𝐴})βˆ€π‘¦ ∈ {𝑆} (π‘₯ ∩ 𝑦) β‰  βˆ…)
37 simp1 1136 . . . . . . . . 9 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
384clsss3 22570 . . . . . . . . . . . . 13 ((𝐽 ∈ Top ∧ 𝑆 βŠ† βˆͺ 𝐽) β†’ ((clsβ€˜π½)β€˜π‘†) βŠ† βˆͺ 𝐽)
393, 25, 38syl2anc 584 . . . . . . . . . . . 12 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ ((clsβ€˜π½)β€˜π‘†) βŠ† βˆͺ 𝐽)
4039, 26sseldd 3983 . . . . . . . . . . 11 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ 𝐴 ∈ βˆͺ 𝐽)
4140, 8eleqtrrd 2836 . . . . . . . . . 10 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ 𝐴 ∈ 𝑋)
4241snssd 4812 . . . . . . . . 9 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ {𝐴} βŠ† 𝑋)
43 snnzg 4778 . . . . . . . . . 10 (𝐴 ∈ ((clsβ€˜π½)β€˜π‘†) β†’ {𝐴} β‰  βˆ…)
44433ad2ant3 1135 . . . . . . . . 9 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ {𝐴} β‰  βˆ…)
45 neifil 23391 . . . . . . . . 9 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ {𝐴} βŠ† 𝑋 ∧ {𝐴} β‰  βˆ…) β†’ ((neiβ€˜π½)β€˜{𝐴}) ∈ (Filβ€˜π‘‹))
4637, 42, 44, 45syl3anc 1371 . . . . . . . 8 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ ((neiβ€˜π½)β€˜{𝐴}) ∈ (Filβ€˜π‘‹))
47 filfbas 23359 . . . . . . . 8 (((neiβ€˜π½)β€˜{𝐴}) ∈ (Filβ€˜π‘‹) β†’ ((neiβ€˜π½)β€˜{𝐴}) ∈ (fBasβ€˜π‘‹))
4846, 47syl 17 . . . . . . 7 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ ((neiβ€˜π½)β€˜{𝐴}) ∈ (fBasβ€˜π‘‹))
49 ne0i 4334 . . . . . . . . . . 11 (𝐴 ∈ ((clsβ€˜π½)β€˜π‘†) β†’ ((clsβ€˜π½)β€˜π‘†) β‰  βˆ…)
50493ad2ant3 1135 . . . . . . . . . 10 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ ((clsβ€˜π½)β€˜π‘†) β‰  βˆ…)
51 cls0 22591 . . . . . . . . . . 11 (𝐽 ∈ Top β†’ ((clsβ€˜π½)β€˜βˆ…) = βˆ…)
523, 51syl 17 . . . . . . . . . 10 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ ((clsβ€˜π½)β€˜βˆ…) = βˆ…)
5350, 52neeqtrrd 3015 . . . . . . . . 9 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ ((clsβ€˜π½)β€˜π‘†) β‰  ((clsβ€˜π½)β€˜βˆ…))
54 fveq2 6891 . . . . . . . . . 10 (𝑆 = βˆ… β†’ ((clsβ€˜π½)β€˜π‘†) = ((clsβ€˜π½)β€˜βˆ…))
5554necon3i 2973 . . . . . . . . 9 (((clsβ€˜π½)β€˜π‘†) β‰  ((clsβ€˜π½)β€˜βˆ…) β†’ 𝑆 β‰  βˆ…)
5653, 55syl 17 . . . . . . . 8 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ 𝑆 β‰  βˆ…)
57 snfbas 23377 . . . . . . . 8 ((𝑆 βŠ† 𝑋 ∧ 𝑆 β‰  βˆ… ∧ 𝑋 ∈ 𝐽) β†’ {𝑆} ∈ (fBasβ€˜π‘‹))
5820, 56, 19, 57syl3anc 1371 . . . . . . 7 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ {𝑆} ∈ (fBasβ€˜π‘‹))
59 fbunfip 23380 . . . . . . 7 ((((neiβ€˜π½)β€˜{𝐴}) ∈ (fBasβ€˜π‘‹) ∧ {𝑆} ∈ (fBasβ€˜π‘‹)) β†’ (Β¬ βˆ… ∈ (fiβ€˜(((neiβ€˜π½)β€˜{𝐴}) βˆͺ {𝑆})) ↔ βˆ€π‘₯ ∈ ((neiβ€˜π½)β€˜{𝐴})βˆ€π‘¦ ∈ {𝑆} (π‘₯ ∩ 𝑦) β‰  βˆ…))
6048, 58, 59syl2anc 584 . . . . . 6 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ (Β¬ βˆ… ∈ (fiβ€˜(((neiβ€˜π½)β€˜{𝐴}) βˆͺ {𝑆})) ↔ βˆ€π‘₯ ∈ ((neiβ€˜π½)β€˜{𝐴})βˆ€π‘¦ ∈ {𝑆} (π‘₯ ∩ 𝑦) β‰  βˆ…))
6136, 60mpbird 256 . . . . 5 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ Β¬ βˆ… ∈ (fiβ€˜(((neiβ€˜π½)β€˜{𝐴}) βˆͺ {𝑆})))
62 fsubbas 23378 . . . . . 6 (𝑋 ∈ 𝐽 β†’ ((fiβ€˜(((neiβ€˜π½)β€˜{𝐴}) βˆͺ {𝑆})) ∈ (fBasβ€˜π‘‹) ↔ ((((neiβ€˜π½)β€˜{𝐴}) βˆͺ {𝑆}) βŠ† 𝒫 𝑋 ∧ (((neiβ€˜π½)β€˜{𝐴}) βˆͺ {𝑆}) β‰  βˆ… ∧ Β¬ βˆ… ∈ (fiβ€˜(((neiβ€˜π½)β€˜{𝐴}) βˆͺ {𝑆})))))
6319, 62syl 17 . . . . 5 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ ((fiβ€˜(((neiβ€˜π½)β€˜{𝐴}) βˆͺ {𝑆})) ∈ (fBasβ€˜π‘‹) ↔ ((((neiβ€˜π½)β€˜{𝐴}) βˆͺ {𝑆}) βŠ† 𝒫 𝑋 ∧ (((neiβ€˜π½)β€˜{𝐴}) βˆͺ {𝑆}) β‰  βˆ… ∧ Β¬ βˆ… ∈ (fiβ€˜(((neiβ€˜π½)β€˜{𝐴}) βˆͺ {𝑆})))))
6417, 24, 61, 63mpbir3and 1342 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ (fiβ€˜(((neiβ€˜π½)β€˜{𝐴}) βˆͺ {𝑆})) ∈ (fBasβ€˜π‘‹))
65 fgcl 23389 . . . 4 ((fiβ€˜(((neiβ€˜π½)β€˜{𝐴}) βˆͺ {𝑆})) ∈ (fBasβ€˜π‘‹) β†’ (𝑋filGen(fiβ€˜(((neiβ€˜π½)β€˜{𝐴}) βˆͺ {𝑆}))) ∈ (Filβ€˜π‘‹))
6664, 65syl 17 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ (𝑋filGen(fiβ€˜(((neiβ€˜π½)β€˜{𝐴}) βˆͺ {𝑆}))) ∈ (Filβ€˜π‘‹))
671, 66eqeltrid 2837 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ 𝐹 ∈ (Filβ€˜π‘‹))
68 fvex 6904 . . . . . 6 ((neiβ€˜π½)β€˜{𝐴}) ∈ V
69 snex 5431 . . . . . 6 {𝑆} ∈ V
7068, 69unex 7735 . . . . 5 (((neiβ€˜π½)β€˜{𝐴}) βˆͺ {𝑆}) ∈ V
71 ssfii 9416 . . . . 5 ((((neiβ€˜π½)β€˜{𝐴}) βˆͺ {𝑆}) ∈ V β†’ (((neiβ€˜π½)β€˜{𝐴}) βˆͺ {𝑆}) βŠ† (fiβ€˜(((neiβ€˜π½)β€˜{𝐴}) βˆͺ {𝑆})))
7270, 71ax-mp 5 . . . 4 (((neiβ€˜π½)β€˜{𝐴}) βˆͺ {𝑆}) βŠ† (fiβ€˜(((neiβ€˜π½)β€˜{𝐴}) βˆͺ {𝑆}))
73 ssfg 23383 . . . . . 6 ((fiβ€˜(((neiβ€˜π½)β€˜{𝐴}) βˆͺ {𝑆})) ∈ (fBasβ€˜π‘‹) β†’ (fiβ€˜(((neiβ€˜π½)β€˜{𝐴}) βˆͺ {𝑆})) βŠ† (𝑋filGen(fiβ€˜(((neiβ€˜π½)β€˜{𝐴}) βˆͺ {𝑆}))))
7464, 73syl 17 . . . . 5 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ (fiβ€˜(((neiβ€˜π½)β€˜{𝐴}) βˆͺ {𝑆})) βŠ† (𝑋filGen(fiβ€˜(((neiβ€˜π½)β€˜{𝐴}) βˆͺ {𝑆}))))
7574, 1sseqtrrdi 4033 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ (fiβ€˜(((neiβ€˜π½)β€˜{𝐴}) βˆͺ {𝑆})) βŠ† 𝐹)
7672, 75sstrid 3993 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ (((neiβ€˜π½)β€˜{𝐴}) βˆͺ {𝑆}) βŠ† 𝐹)
77 snssg 4787 . . . . 5 (𝑆 ∈ V β†’ (𝑆 ∈ (((neiβ€˜π½)β€˜{𝐴}) βˆͺ {𝑆}) ↔ {𝑆} βŠ† (((neiβ€˜π½)β€˜{𝐴}) βˆͺ {𝑆})))
7821, 77syl 17 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ (𝑆 ∈ (((neiβ€˜π½)β€˜{𝐴}) βˆͺ {𝑆}) ↔ {𝑆} βŠ† (((neiβ€˜π½)β€˜{𝐴}) βˆͺ {𝑆})))
7918, 78mpbiri 257 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ 𝑆 ∈ (((neiβ€˜π½)β€˜{𝐴}) βˆͺ {𝑆}))
8076, 79sseldd 3983 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ 𝑆 ∈ 𝐹)
8176unssad 4187 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ ((neiβ€˜π½)β€˜{𝐴}) βŠ† 𝐹)
82 elflim 23482 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹)) β†’ (𝐴 ∈ (𝐽 fLim 𝐹) ↔ (𝐴 ∈ 𝑋 ∧ ((neiβ€˜π½)β€˜{𝐴}) βŠ† 𝐹)))
8337, 67, 82syl2anc 584 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ (𝐴 ∈ (𝐽 fLim 𝐹) ↔ (𝐴 ∈ 𝑋 ∧ ((neiβ€˜π½)β€˜{𝐴}) βŠ† 𝐹)))
8441, 81, 83mpbir2and 711 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ 𝐴 ∈ (𝐽 fLim 𝐹))
8567, 80, 843jca 1128 1 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋 ∧ 𝐴 ∈ ((clsβ€˜π½)β€˜π‘†)) β†’ (𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝑆 ∈ 𝐹 ∧ 𝐴 ∈ (𝐽 fLim 𝐹)))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   β‰  wne 2940  βˆ€wral 3061  Vcvv 3474   βˆͺ cun 3946   ∩ cin 3947   βŠ† wss 3948  βˆ…c0 4322  π’« cpw 4602  {csn 4628  βˆͺ cuni 4908  β€˜cfv 6543  (class class class)co 7411  ficfi 9407  fBascfbas 20938  filGencfg 20939  Topctop 22402  TopOnctopon 22419  clsccl 22529  neicnei 22608  Filcfil 23356   fLim cflim 23445
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-iin 5000  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1o 8468  df-er 8705  df-en 8942  df-fin 8945  df-fi 9408  df-fbas 20947  df-fg 20948  df-top 22403  df-topon 22420  df-cld 22530  df-ntr 22531  df-cls 22532  df-nei 22609  df-fil 23357  df-flim 23450
This theorem is referenced by:  flimcls  23496
  Copyright terms: Public domain W3C validator